docs(332): mark Apollonian Phase 5 (T5.1-T5.3) as deferred [-] after reasoning audit

Re-confirmed the deferral is correct, but found and fixed a measurement-selection
bug in the stated justification:

- The original deferral cited '+0.0247 = hand - Haar at k=8/tau=0.5' as the
  'achievable ceiling over Haar'. This number is from the narrow probe signal,
  which the benchmark doc itself classifies as a 'narrow-low-frequency artifact,
  not representative of realistic PDE spectral content'.
- On the broadband PDE-like signal (the fair test), hand-crafted is NOT the
  upper bound - DCT-log beats it by +0.17. So the 'Apollonian sits between Haar
  and hand' logic chain breaks on broadband.

Conclusion survives the bug (defer is still correct), for three independent reasons:
1. On broadband, the winning fixed basis is DCT-log (+0.34, spectral family),
   not Haar (+0.16, localized family). Apollonian is in the localized family,
   which is currently -0.18 behind on broadband.
2. The k>=16 rank-saturation elbow is signal-independent (curse of dimensionality).
3. No off-the-shelf d=64 Apollonian harmonic constructor exists (research-grade).

Verification (re-ran cargo test -p katgpt-core --features funcattn_structured_basis
--test funcattn_structured_basis_g1): all 4 tests pass, numbers reproduce
bit-identically (+0.5900/+0.5652/+0.4806/+0.3379 at tau=0.5/k=8).

T5.1-T5.3 marked [-] (deferred) with corrected justification. Original
'revisit if +0.02 cos gap is blocking' criterion withdrawn (signal-specific).

Author: katopz

.benchmarks/332_structured_basis_goat_and_k_sweep.md

@@ -166,11 +166,11 @@ Do NOT use `make_dct_log_basis` for **narrow low-frequency** transport tasks (th
 The plan gated Apollonian harmonics on a simpler multi-scale basis (Haar-packet) passing first. **Haar-packet passed at k≤8, τ=0.5** — the Apollonian-surrogate door is ajar, not closed.
 
 However:
-- Haar's win is narrow (+0.0846 at k=8, vs the +0.1093 achievable). Apollonian's richer geometry would need to beat Haar by a meaningful margin to justify the implementation cost.
-- The loss at k≥16 and τ=0.1 means Apollonian would face the same rank-saturation and sharp-sigmoid problems.
+- Haar's win on the narrow probe signal is narrow (+0.0846 at k=8, vs the +0.1093 of hand-crafted there), but **on the broadband PDE-like signal (the fair test), the winning fixed basis is DCT-log (+0.34), not Haar (+0.16)**. Apollonian's claimed advantage is richer localized multi-scale geometry (the Haar family) — but on broadband signals the localized family already loses to the spectral family by −0.18. So Apollonian's headroom over the *current best* fixed basis is bounded by Haar's gap to DCT-log, not by hand's gap to Haar. (The earlier "+0.0247 ceiling" framing was withdrawn after a 2026-06-28 audit: that number is narrow-probe-signal-specific and the benchmark itself classifies that signal as a non-representative artifact.)
+- The loss at k≥16 and τ=0.1 means Apollonian would face the same rank-saturation and sharp-sigmoid problems. The k≥16 elbow is signal-independent (curse of dimensionality at k≈d/4).
 - The k-sweep elbow at k=16 suggests the maximum addressable regime for any fixed structured basis is k∈[4, 16] — a small window.
 
-**Recommendation for Phase 5**: do NOT implement true d-dimensional Apollonian harmonics yet. The Phase 2 result shows the achievable gain is bounded (+0.08 to +0.11 cos), localized to small k, and already mostly captured by Haar. Apollonian's extra geometric richness is unlikely to clear the implementation-cost bar. Revisit only if a concrete use case emerges where the +0.02 cos gap between Haar and the achievable bound is the blocking factor.
+**Recommendation for Phase 5**: do NOT implement true d-dimensional Apollonian harmonics yet. On the realistic broadband regime, a fixed *spectral* basis (DCT-log) already captures +0.34 cos over random — the gain is banked. Apollonian (localized family) would need to beat DCT-log, not Haar, to justify its research-grade implementation cost, and the localized family is currently −0.18 behind. Revisit only if a concrete use case emerges where (a) the task is in k∈[4,8] × τ≥0.5, AND (b) the signal is localized-multi-scale (Haar-family territory, not broadband-spectral), AND (c) the gap to the achievable bound is the blocking factor.
 
 ---
 

.plans/332_structured_basis_selection_for_funcattn.md

@@ -140,9 +140,11 @@ If DCT-log or Haar-packet passes G1+G2, THEN we justify the harder work of imple
 2. **True d-dim Apollonian harmonics are a research project.** No off-the-shelf implementation exists for d=64. Implementing them requires either (a) 2D/3D projection (lossy), (b) construction from scratch via the Apollonian group, or (c) a different non-Euclidean embedding.
 3. **Phase 5 is gated on Phase 2.** Only pursue Apollonian harmonics if a simpler principled multi-scale basis (Haar-packet) already proves the concept.
 
-- [ ] T5.1 (GATED on Phase 2 PASS) Literature search: Apollonian harmonic decompositions in d > 3 dimensions (arxiv search via jina) — **DEFERRED 2026-06-26:** Phase 2 produced a partial PASS (Haar at τ=0.5/k≤8) but the achievable gain over Haar is narrow (+0.0247 = hand − Haar at k=8/τ=0.5), and Haar already loses at k≥16. Apollonian's extra geometric richness is unlikely to clear the implementation-cost bar.
-- [ ] T5.2 (GATED) Prototype Apollonian harmonic basis constructor (likely via 2D projection or surrogate) — **DEFERRED.**
-- [ ] T5.3 (GATED) Benchmark Apollonian vs Haar-packet vs DCT-log — does Apollonian's richer geometry beat the simpler multi-scale bases? — **DEFERRED.** Revisit only if a concrete use case emerges where the +0.02 cos gap between Haar and the achievable bound is the blocking factor.
+- [-] T5.1 (GATED on Phase 2 PASS) Literature search: Apollonian harmonic decompositions in d > 3 dimensions (arxiv search via jina) — **DEFERRED 2026-06-26 (re-confirmed 2026-06-28 after reasoning audit):** Phase 2 produced a partial PASS (Haar at τ=0.5/k≤8 on the narrow probe signal), and the per-basis gate on the broadband PDE-like signal (the fair test) already promoted DCT-log + Haar-packet to DEFAULT-ON.
+  - **Original (flawed) justification cited +0.0247 = hand − Haar at k=8/τ=0.5 as the "achievable ceiling over Haar".** Audit (2026-06-28) found this is a measurement-selection bug: that number comes from the narrow probe signal, which the benchmark doc itself classifies as a "narrow-low-frequency artifact, not representative of realistic PDE spectral content". On the broadband PDE-like signal, hand-crafted is *not* the upper bound (DCT-log beats it by +0.17), so the "Apollonian sits between Haar and hand" logic chain breaks there.
+  - **Corrected justification (conclusion unchanged):** on the broadband signal the winning fixed basis is the *spectral* DCT-log (+0.34), not the *localized* Haar (+0.16). Apollonian's claimed advantage over Haar is richer localized multi-scale geometry — but on broadband signals the localized family already loses to the spectral family by −0.18, so Apollonian's headroom over the current best is bounded by Haar's gap to DCT-log, not by hand's gap to Haar. Additionally: the k≥16 rank-saturation elbow is signal-independent (curse of dimensionality, k≈d/4), so the addressable regime stays k∈[4,8] × τ≥0.5 for any fixed basis; and no off-the-shelf d=64 Apollonian harmonic constructor exists (T5.2 is research-grade). The modelless gain was already banked by the 2026-06-26 default-on promotion.
+- [-] T5.2 (GATED) Prototype Apollonian harmonic basis constructor (likely via 2D projection or surrogate) — **DEFERRED.** Research-grade cost (no off-the-shelf d=64 constructor) not justified by the headroom analysis in T5.1.
+- [-] T5.3 (GATED) Benchmark Apollonian vs Haar-packet vs DCT-log — does Apollonian's richer geometry beat the simpler multi-scale bases? — **DEFERRED.** Revisit only if a concrete use case emerges where (a) the task is in the k∈[4,8] × τ≥0.5 regime, AND (b) the signal is localized-multi-scale (Haar-family territory, not broadband-spectral where DCT-log already wins), AND (c) the gap to the achievable bound is the blocking factor. The original "revisit if the +0.02 cos gap is blocking" criterion is withdrawn — that number is signal-specific and not a universal ceiling.
 
 ---